Theoretical evaluation of the temperature dependence of magnetic anisotropy constants of Nd₂Fe₁₄B: Effects of exchange field and crystal field strength

Abstract

To identify the possible mechanism of coercivity (H c ) degradation of Nd-Fe-B sintered magnets, we study the roles of the exchange field acting on the 4f electrons in Nd ions and theoretically investigate how the variation of the exchange field affects the values of the magnetic anisotropy constants K 1 and K 2 . We find that, with decreasing exchange field strength, both values decrease as a result of the lower asphericity of the 4f electron cloud, indicating that the local anisotropy constants might become small around the grain boundaries where the exchange fields are decreased owing to the smaller coordination number.Nd-Fe-B sintered magnets [1][2][3] have the largest maximum energy product among the current magnets and have been widely used for magnetic devices such as voice coil motors in magnetic recording systems. Recently, because of the rapidly growing interest in electric vehicles, much effort has been made to suppress the degradation of the coercivity (H c ) of Nd-Fe-B magnets. However, from an industrial viewpoint, reduction in the usage of Dy is strongly desired, because Dy is a rare metal and the magnetization of the Nd-Fe-B magnets decreases by substituting Dy with Nd owing to the antiparallel coupling between Dy and Fe moments. Realizing Dy-free high-performance Nd-Fe-B magnets requires a further increase of H c in the Nd-Fe-B system by microstructure optimization, [4][5][6][7][8][9][10][11] and therefore, establishing the microscopic foundation for the coercivity mechanism is desired. From a theoretical viewpoint, many works [12][13][14][15][16] have focused on the change of magnetic anisotropy constants around the grain boundary surfaces as a result of the stresses, defects, and change of spatial symmetry. In addition, micromagnetic model calculations have shown that the surface c-plane anisotropy can drastically decrease the coercivity. 17) For these reasons, evaluation of the local anisotropy constants around the grain boundaries and determining their temperature dependence are important to investigate the degradation of the coercivity.With regard to the magnetic anisotropy of rare earth (RE) transition metal compounds, it is believed that the 4f electrons in RE ions are responsible for the main part of the magnetic anisotropy and that the crystalline electric field (CEF) acting on the 4f electrons dominates this property. 18) Under the assumption that the exchange field on the 4f electrons is strong enough, by using the CEF parameter A 0 2 , the leading anisotropy constant K 1 can be approximately described by K 1 = −3J(J −1)α r 2 A 0 2 N R , where α is the Stevens factor, J is the total angular momentum, r 2 is the average ofr 2 over the radial wave function of the 4f electrons, and N R is the density of RE ions. Note that the CEF parameter

Calculating temperature-dependent properties of Nd2Fe14B permanent magnets by atomistic spin model simulations

“…We can see that below 150 K, K eff 1 is negative and both K eff 2 and K eff 3 play a critical role, agreeing Fig. 3(c) agrees reasonably with the previous experimental measurement [42,43,49] and theoretical calculations [24,50].…”

Non-collinearity Effects on Magnetocrystalline Anisotropy for R2Fe14B Magnets

“…In the lowtemperature range below the spin-reorientation transition (SRT) temperature (T ∼ 130 K [45]), these compounds exhibit complex behavior as shown by the lines for T = 0, 100, and 135 K. Above the SRT temperature, we can observe that ∆Θ < 0 because M Nd (θ, φ; T ) tends to naively orient along the +z-axis, and that |∆Θ| monotonically decreases with increasing temperature. In Nd 2 Fe 14 B magnets, |∆Θ| has an extremely low value over the entire temperature range, and thus we conclude that the NCE is negligibly small as assumed in our previous studies [24][25][26].…”

“…In our previous studies on the MA of Nd 2 Fe 14 B magnets [24][25][26], the total magnetization was assumed to be collinear to the Fe magnetization. However, this assumption raises a serious error in evaluations for the MA of the R 2 Fe 14 B magnet, the R magnetization of which is highly non-collinear to its Fe magnetization.…”

Power law analysis for temperature dependence of magnetocrystalline anisotropy constants of Nd2Fe14B magnets

“…Here, we emphasize that the common parameters in the two comparisons are used in µ Nd (T ), and particularly that the value of H Nd = 350 K is reasonable because the strength of the EXF strongly influences the plateau of the MACs appearing in the low-temperature range. 9 In addition, we must consider the non-collinearity between the R and Fe moments. Although the MACs of R =Nd obey the present power law well, this is not the case for magnets with a high non-collinearity.…”

“…For example, Yamada et al showed through their systematic analysis of the magnetization process for the R 2 Fe 14 B series that parameters representing crystalline electric fields (CEFs) and effective exchange fields (EXFs) at R sites do not depend significantly on R. 8 Furthermore, we have previously confirmed that an effective Hamiltonian using the parameters suitably describes the experimental temperature dependence of the magnetocrystalline anisotropy constants (MACs) of Nd 2 Fe 14 B magnets. 9,10 However, there remains a gap between our microscopic picture and the macroscopic aspects of MA since previous theoretical studies on the temperature dependence of MA for R 2 Fe 14 B magnets are based on complete diagonalization of the Hamiltonian 9,10 or numerical methods such as the Monte Carlo method. 4,11,12 An effective means of bridging this gap is to develop a power-law theory 5,6,[13][14][15][16][17][18] to establish the relationship between MACs and magnetization.…”